Introduction to Algebra

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• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities
• It provides a framework for solving equations and inequalities
• It explores relationships between variables

Fundamental Concepts

• Variable: A symbol (usually a letter) that represents an unknown or changing quantity
• Constant: A fixed value that does not change
• Coefficient: A number that multiplies a variable
• Expression: A combination of variables, constants, and operations (addition, subtraction, multiplication, division, exponentiation)
• Equation: A statement that two expressions are equal, connected by an equals sign (=)
• Term: A single number or variable, or numbers and variables multiplied together

Operations

• Addition (+): Combining two or more terms
• Subtraction (-): Finding the difference between two terms
• Multiplication (× or *): Repeated addition of a term
• Division (÷ or /): Splitting a term into equal parts
• Exponentiation (^): Raising a term to a power (repeated multiplication)
• Order of Operations: PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction)

Solving Equations

• Isolate the variable: Manipulating the equation to get the variable alone on one side of the equals sign
• Use inverse operations: Perform the opposite operation to undo the operation being applied to the variable
• Addition and Subtraction Property of Equality: Adding or subtracting the same value from both sides of an equation maintains equality
• Multiplication and Division Property of Equality: Multiplying or dividing both sides of an equation by the same non-zero value maintains equality
• Distributive Property: Multiplying a term by an expression in parentheses (a(b+c) = ab + ac)
• Combining Like Terms: Simplifying an expression by adding or subtracting terms with the same variable and exponent

Linear Equations

• Standard Form: ax + b = c, where a, b, and c are constants and x is the variable
• Solving Linear Equations: Use inverse operations to isolate the variable
• Equations with Variables on Both Sides: Simplify each side, then use inverse operations to move all variable terms to one side and constant terms to the other
• No Solution: When solving, the variable is eliminated and the resulting statement is false (e.g., 0 = 1)
• Infinite Solutions: When solving, the variable is eliminated and the resulting statement is true (e.g., 0 = 0)

Inequalities

• Inequality Symbols:
  • <: Less than
  • : Greater than
  • ≤: Less than or equal to
  • ≥: Greater than or equal to
• Solving Inequalities: Similar to solving equations, but with one key difference:
  • Multiplying or dividing both sides by a negative number reverses the inequality sign
• Graphing Inequalities: Representing the solution set on a number line using open or closed circles and arrows
• Interval Notation: Expressing the solution set as an interval using parentheses and brackets (e.g., (a, b), [a, b], (a, ∞))

Systems of Equations

• Two or more equations with the same variables
• Goal: To find the values of the variables that satisfy all equations simultaneously
• Methods for Solving:
  • Graphing: Plotting the equations and finding the point of intersection
  • Substitution: Solving one equation for one variable and substituting that expression into the other equation
  • Elimination (Addition/Subtraction): Manipulating the equations so that the coefficients of one variable are opposites, then adding the equations to eliminate that variable
• Types of Solutions:
  • One Solution: The lines intersect at a single point
  • No Solution: The lines are parallel and do not intersect
  • Infinite Solutions: The lines are the same (coincident)

Polynomials

• An expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents
• Terms: Parts of an expression separated by + or - signs
• Degree of a term: The exponent of the variable
• Degree of a polynomial: The highest degree of any term in the polynomial
• Types of Polynomials:
  • Monomial: One term (e.g., 5x)
  • Binomial: Two terms (e.g., 2x + 3)
  • Trinomial: Three terms (e.g., x² - 4x + 7)
• Operations with Polynomials:
  • Adding and Subtracting: Combine like terms
  • Multiplying: Use the distributive property or FOIL method
  • Dividing: Polynomial long division or synthetic division

Factoring Polynomials

• Expressing a polynomial as a product of simpler polynomials or monomials
• Common Factoring Techniques:
  • Greatest Common Factor (GCF): Factoring out the largest factor that divides all terms
  • Difference of Squares: a² - b² = (a + b)(a - b)
  • Perfect Square Trinomial: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²
  • Factoring by Grouping: Grouping terms and factoring out common factors from each group
  • Factoring Quadratics: Finding two binomials that multiply to give the quadratic expression

Quadratic Equations

• Standard Form: ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0
• Solving Quadratic Equations:
  • Factoring: Factor the quadratic expression and set each factor equal to zero
  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
  • Completing the Square: Manipulating the equation to create a perfect square trinomial
• Discriminant: The expression b² - 4ac in the quadratic formula, which determines the number and type of solutions:
  • b² - 4ac > 0: Two distinct real solutions
  • b² - 4ac = 0: One real solution (a repeated root)
  • b² - 4ac < 0: Two complex solutions

Rational Expressions

• A fraction where the numerator and denominator are polynomials
• Simplifying Rational Expressions: Factoring the numerator and denominator and canceling common factors
• Operations with Rational Expressions:
  • Multiplying: Multiply numerators and denominators
  • Dividing: Multiply by the reciprocal of the divisor
  • Adding and Subtracting: Find a common denominator and combine numerators
• Solving Rational Equations: Multiplying both sides by the least common denominator (LCD) to eliminate fractions

Exponents and Radicals

• Exponent Rules:
  • Product of Powers: a^m * a^n = a^(m+n)
  • Quotient of Powers: a^m / a^n = a^(m-n)
  • Power of a Power: (a^m)^n = a^(m*n)
  • Power of a Product: (ab)^n = a^n * b^n
  • Power of a Quotient: (a/b)^n = a^n / b^n
  • Zero Exponent: a^0 = 1 (if a ≠ 0)
  • Negative Exponent: a^(-n) = 1 / a^n
• Radicals:
  • √a: The principal square root of a
  • index√a: The nth root of a
  • Simplifying Radicals: Factoring out perfect squares/cubes, etc.
• Rationalizing the Denominator: Eliminating radicals from the denominator of a fraction
• Relationship between Exponents and Radicals: a^(m/n) = (index√a)^m = index√(a^m)

Functions

• A relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output
• Notation: f(x) (read as "f of x") represents the output of the function f for the input x
• Types of Functions:
  • Linear Functions: f(x) = mx + b (straight line)
  • Quadratic Functions: f(x) = ax² + bx + c (parabola)
  • Exponential Functions: f(x) = a^x
  • Logarithmic Functions: f(x) = log_a(x)
• Domain: The set of all possible input values (x-values)
• Range: The set of all possible output values (y-values)
• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph more than once